Momentum and Impulse

Linear Momentum

Momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It is a vector quantity, meaning it has both magnitude and direction.

• Definition: The linear momentum p of an object is given by the product of its mass and velocity.

• Formula:

• Units: kg·m/s

• Vector Nature: Both p and v are vectors; always consider their components when solving problems.

Impulse and Change in Momentum

Impulse quantifies the effect of a force acting over a time interval, resulting in a change in momentum.

• Impulse (J): The product of average force and the time interval during which the force acts.

• Formula:

• Units: N·s or kg·m/s

• Key Point: Impulse equals the change in momentum.

Rotational Kinematics

Angular Position, Velocity, and Acceleration

Rotational motion describes the movement of objects around a fixed axis. The angular analogs of position, velocity, and acceleration are used.

• Angular Position (θ): Measured in radians (rad).

• Angular Velocity (ω): Rate of change of angular position, measured in rad/s.

• Angular Acceleration (α): Rate of change of angular velocity, measured in rad/s2.

• Conversions: rad = 360°, rad = 180°

Arc Length and Tangential Quantities

• Arc Length (s):

• Radius (r): Distance from axis of rotation to the point of interest (m).

• Tangential Velocity (v):

• Tangential Acceleration (a_{tan}):

• Note: For objects rolling without slipping,

Rotational Dynamics

Moment of Inertia

The moment of inertia quantifies how mass is distributed relative to the axis of rotation and determines an object's resistance to changes in rotational motion.

• Definition: (for discrete masses)

• Units: kg·m2

• Point Mass:

Rotational Kinetic Energy

• Formula:

• Total Kinetic Energy (Rigid Body):

• Where: is the center of mass velocity.

Torque

Torque is the rotational equivalent of force; it measures the tendency of a force to rotate an object about an axis.

• Definition:

• Units: N·m

• Moment Arm (l): The perpendicular distance from the axis of rotation to the line of action of the force.

Newton's Second Law for Rotation

• Formula:

• Key Point: The net torque on a body is equal to the product of its moment of inertia and angular acceleration.

• For rolling without slipping:

Work and Power in Rotational Motion

• Rotational Work:

• Rotational Power (average):

Angular Momentum

• Rigid Body:

• Point Mass:

• Conservation: Angular momentum is conserved in the absence of external torques.

Energy in Rotational Motion

Gravitational Potential Energy

• Formula:

• Where: is the vertical height of the center of mass relative to a reference point ().

• Common Simplification:

Static Equilibrium

Conditions for Equilibrium

• Translational Equilibrium:

• Rotational Equilibrium:

• Application: Choose an axis of rotation to simplify torque calculations. Forces that pass through the axis produce no torque.

Summary Table: Key Rotational Quantities

Example Application

• Example: A solid disk of mass 2 kg and radius 0.5 m rotates at 4 rad/s. Find its rotational kinetic energy.

• Solution: For a solid disk,

• Plug in values: kg·m2

• J

Additional info: Some context and clarifications were added for completeness and academic clarity.